NEET Chemistry The Solid State Questions Solved

A metallic element crystallizes into a lattice containing a sequence of layers of ABABAB ............ Any packing of spheres leaves out voids in the lattice. The percentage by volume of empty space of this is

(1) 26%

(2) 21%

(3) 18%

(4) 16 %

(1) The hexagonal base consists of six equilateral triangles, each with side 2r and altitude 2r sin 60°.

Hence, area of base = 6 12(2r)(2rsin60o)=63.r2

The height of the hexagonal is twice the distance between closest packed layers.

The latter can be determined to a face centred cubic lattice with unit cell length a. In such a lattice, the distance between closest packed layers is one third of the body diagonal, i.e. 3a3

Hence  Height (h)=23a3=2a3

Now, in the face centred lattice, atoms touch one another along the face diagonal,

Thus, 4r=2.a

With this, the height of hexagonal becomes :  Height (h)=234r2=423.r

Volume of hexagonal unit is, V = (base area) × (height) =(63r2)423.r=242.r3

In one hexagonal unit cell, there are 6 atoms as described below :

• 3 atoms in the central layer which exclusively belong to the unit cell.

• 1 atom from the centre of the base. There are two atoms of this type and each is shared between two hexagonal unit cells.

• 2 atoms from the corners. There are 12 such atoms and each is shared amongst six hexagonal unit cells.

Now, the volume occupied by atoms = 643πr3

Fraction of volume occupied by atoms =Volume occupied by atomsVolume of hexagonal unit cell=643πr3242.r3=π/32=0.74.

Fraction of empty space = (1.000.74)=0.26

Percentage of empty space = 26%   

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